Issue 29

J. Toti et alii, Frattura ed Integrità Strutturale, 29 (2014) 166-177; DOI: 10.3221/IGF-ESIS.29.15

2

Numerical data Damage model

1.8

1.6

A

1.4

1

3

5

B 7

1.2

1

0.8 F [kN]

0.6

0.4

0.2

0

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

v [mm] 8 Figure 6 : Cyclic test: comparison between numerical and experimental results. 2 4 6

_________________ S T R E S S 4 D t

D t

0.00E+00 1.00E-01 2.00E-01 3.00E-01 4.00E-01 5.00E-01 6.00E-01 7.00E-01 8.00E-01 9.00E-01 1.00E+00 9.66E-01 0.00E+00

0.00E+00 1.00E-01 2.00E-01 3.00E-01 4.00E-01 5.00E-01 6.00E-01 7.00E-01 8.00E-01 9.00E-01 1.00E+00 9.95E-01 0.00E+00

b)

a)

Figure 7 : Tensile damage map: a) time step A; b) time step B;

Loading history time step 0 1 2 3 4 5 6 7 8 [mm] 0 0.38 0.04 0.42 0.07 0.45 0.1 0.9 0.24 v Table 4 : Loading history for the displacement v

Dynamic test Numerical analyses are conducted to investigate the damage propagation induced by a sinusoidal synchronous motion applied to all nodes of base     0 0 sin 2 g u t u f t   , where 0 u and 0 f indicate the displacement amplitude and the frequency of the harmonic imposed displacement, respectively. The propagation of the damage in the arch is analyzed for frequency ratio 0 1 f f   equal to 0.7 and 1.3. For each frequency ratio, two values of the displacement amplitude are considered: 0 0.10 mm u  and 0 0.11mm u  . The equations of motion are integrated until the time 5 s is reached, with a time step equal to 0.001 s (approximately 1/14 of the second natural period of vibration) in order to correctly take into account the dynamics of the first two modes. The comparison between the damage model and the elastic model is provided in term of time histories of the displacement v evaluated at the extrados of the key-section of the arch. The effect of damage on the dynamic response is analyzed through the Fourier spectral analyses. Fig. 8 and Fig. 9 show the displacement response and Fourier spectra for all the considered forcing functions, adopting both elastic and the proposed damage model. Observing these figures, it can be remarked that: the arch collapses before 5 s for 0.7   with 0 0.11mm u  and for 1.3   with 0 0.10 mm u  or ; the development of the damage in the nonlinear model leads, away from the failure mechanism, only to a decrease in time of the natural frequencies, while the frequency contents become richer for all cases close to the collapse condition; the reduction of the frequency contents became faster with increasing the forcing displacement amplitude. In particular, for 0.7   with 0 0.11mm u  at the time equal to 1.5 s the first natural frequency of the damaged system is approaching the lower forcing one, the arch goes in resonance following its first modal shape and, as consequence, collapses. On the contrary, for the other case, characterized by lower forcing displacement amplitudes, the structural 0 u  0.11mm

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